Question: $ B = \left[\begin{array}{rrr}1 & 4 & 4 \\ -1 & 3 & -2\end{array}\right]$ $ F = \left[\begin{array}{rr}-2 & 2 \\ 3 & 0 \\ 1 & 3\end{array}\right]$ What is $ B F$ ?
Because $ B$ has dimensions $(2\times3)$ and $ F$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ B F = \left[\begin{array}{rrr}{1} & {4} & {4} \\ {-1} & {3} & {-2}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{2} \\ {3} & \color{#DF0030}{0} \\ {1} & \color{#DF0030}{3}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{4}\cdot{3}+{4}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{4}\cdot{3}+{4}\cdot{1} & ? \\ {-1}\cdot{-2}+{3}\cdot{3}+{-2}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{4}\cdot{3}+{4}\cdot{1} & {1}\cdot\color{#DF0030}{2}+{4}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{3} \\ {-1}\cdot{-2}+{3}\cdot{3}+{-2}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{-2}+{4}\cdot{3}+{4}\cdot{1} & {1}\cdot\color{#DF0030}{2}+{4}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{3} \\ {-1}\cdot{-2}+{3}\cdot{3}+{-2}\cdot{1} & {-1}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{0}+{-2}\cdot\color{#DF0030}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}14 & 14 \\ 9 & -8\end{array}\right] $